WASHINGTON—The Senate stepped back Tuesday from the edge of a partisan clash over President Barack Obama's executive-branch nominees, as Republicans agreed to quick consideration of his choices for several senior jobs and Democrats stood down from a threat to limit the minority party's use of filibusters.

The deal paves the way for the confirmations of five of Mr. Obama's executive branch nominees and the expedited consideration of two more named Tuesday. Republicans won the withdrawal of two nominees for the National...